Future Beacon



        
Mathematics

From The Beginning





Introduction


This system is offered as a starting point or foundation for mathematics.   Its distinctive features are these:

        Nothing may contain itself.

        A and B are equal if and only if they have the same meaning.

        The definitions of the terms elements, sets, subsets, empty sets, ordered sets and others are different from those in other systems.

        Definitions may contain references to time such as the terms previously, before, after, and subsequently.

        Sets that contain all of the same elements are coincident, but they are not necessarily equal.

        Axioms are not used.

Mathematics is a direct consequence of our most basic perceptions; therefore, it must not be freely invented.   It must be rigorously derived.   The creation of any given definition is constrained by those meanings established prior to the statement of that definition.

In this work, a statement is not taken to be true unless it is known to be true.   No assumptions are made.   A statement may not be listed as an axiom if it can be shown to be true.   This eliminates axioms.

There are mathematical results that are critically needed by scientists, engineers, programmers, economists, statisticians, and many others who are not principally (if at all) in the business of furthering mathematics itself .   I would place in another category the results valued by mathematicians that are not needed by anybody else.   It will be shown that the mathematical results that are important to non-mathematicians can be obtained without axioms.   The goal is not to create interesting or difficult challenges.   It is to facilitate arithmetic, calculus, statistics, algebra, and all such tools of science.

An entirely empirical mathematics will enhance but not contradict the experience of newcomers.   This can only increase participation.